Yet a poignant scene of her playing “Evergreen” on the guitar was cut from the movie.

On Friday, Netflix is bringing this version of the Grammy-winning song to light with an edition of the movie featuring never-before-seen original footage — the addition of which Ms. Streisand managed, the streaming service said.

Six Streisand television specials dating back to 1965 are also being released. They shine light on her role as a “rule-breaker” of that time, as they are one-woman shows, said Tom Santopietro, author of the biography “The Importance of Being Barbra.”

When Ms. Streisand starred alongside Kris Kristofferson in “A Star Is Born,” it was the third variation of the romance: A club singer, Esther, falls in love with John, an established yet despondent rock star who has started to hit bumps in his career. (The original debuted in 1937; James Mason and Judy Garland starred in a 1954 remake.)

The added guitar version of “Evergreen” was originally to appear in the movie as a soft precursor to what happens later, when the two characters sing the song together and the love story at the heart of the film comes full circle.

But the guitar scene was cut to condense the film; and as the executive producer, Ms. Streisand had the final say on edits. “I learned how to play the guitar and wrote the melody to ‘Evergreen’ for this film — and ended up cutting this scene out,” she said in a statement from her publicist. “Who knew the song would become such a big hit?”

“Evergreen” was a “huge cornerstone” in Ms. Streisand’s recording career, Mr. Santopietro said. It won her two Grammys, including best song, and a Golden Globe for best original song for a motion picture. Ms. Streisand also earned her second Oscar, for co-writing the song (with Mr. Williams), making her the first woman to earn an Academy Award for composing. “Evergreen” also ranked No. 1 on the Billboard Hot 100 for three weeks in March 1977.

Mr. Williams said he had taken home a cassette tape of Ms. Streisand playing the chords, and wrote the lyrics in about a day. (He also worked on six more songs for the movie soundtrack.) As soon as he heard the finished version, he said, “I thought we had a big record.”

Netflix watchers will also find new footage of Ms. Streisand performing a rock ’n’ roll version of the finale, “With One More Look at You/Watch Closely Now.”

“We filmed a lot of rock ’n’ roll kind of footage,” Ms. Streisand explained in a conversation with Jamie Foxx at a Netflix FYSee event on Sunday. But she didn’t have time to properly edit it into the film before the movie’s preview, she said.

And at the preview, she said, the audience loved the seven-minute finale as it was, so she didn’t add in the rock ’n’ roll version. “I thought ‘Uh oh,’ you know, ‘If it ain’t broken, don’t fix it,’” she said.

Mr. Santopietro said: “Even though things aren’t great for women in Hollywood now, they weren’t great then. The fact that she had this sort of power spoke to who she was. It was a personal undertaking for her. This completes her vision.”

A fourth remake of “A Star Is Born” comes out in October, with Lady Gaga alongside Bradley Cooper in a country-music-infused version of the romance.

The six specials being released include the Emmy-winning “My Name Is Barbra” (1965), “Barbra Streisand: The Concert” (1994) and “Barbra Streisand: Timeless” (2001), as well as “Color Me Barbra” (1966), “Barbra Streisand: A Happening in Central Park” (1968) and “Barbra Streisand and Other Musical Instruments” (1973).